Fundamental examples of Reeb spaces of smooth functions defined from two graphs of smooth functions with same asymptotic behaviors

TL;DR

This paper explores the structure of Reeb spaces associated with smooth functions on non-compact manifolds, showing they can be homeomorphic to infinite graphs with ends, extending previous results in the field.

Contribution

It provides new examples of Reeb spaces homeomorphic to infinite graphs with ends for specific smooth functions on non-compact manifolds, building on prior work.

Findings

Reeb spaces can be homeomorphic to infinite graphs with ends.

The study extends understanding of Reeb spaces for functions on non-compact manifolds.

Provides explicit examples of such Reeb spaces.

Abstract

Reeb spaces of (continuous) real-valued functions on (nice) topological spaces are the spaces whose underlying sets consist of all connected components (contours) of their level sets and seen naturally as quotient spaces of the spaces. They are "$1$-dimensional" spaces in various nice cases. They are graphs or graphs with ends for smooth function cases with nice singularities and behaviors. Reeb spaces have been fundamental and important in theory of Morse functions and more general smooth functions and applications to geometry, since the 20th century. We present Reeb spaces homeomorphic to infinite graphs (with ends) for functions on non-compact manifolds with no boundary. This paper is a note on cases previously obtained by the author. More explicitly, we consider a natural smooth map onto the region surrounded by the graphs of two smooth real-valued functions in the plane and its composition with the canonical projection.